The transformation optics approach to cloaking uses a singular change of coordinates, which blows up a point to the region being cloaked. This paper examines a natural regularization, obtained by (1) blowing up a ball of radius rather than a point, and (2) including a well-chosen lossy layer at the inner edge of the cloak. We assess the performance of the resulting near-cloak as the regularization parameter tends to 0, in the context of (Dirichlet and Neumann) boundary measurements for the time-harmonic Helmholtz equation. Since the goal is to achieve cloaking regardless of the content of the cloaked region, we focus on estimates that are uniform with respect to the physical properties of this region. In three space dimensions our regularized construction performs relatively well: the deviation from perfect cloaking is of order . In two space dimensions it does much worse: the deviation is of order 1=jlog j. In addition to proving these estimates, we give numerical examples demonstrating their sharpness. Some authors have argued that perfect cloaking can be achieved without losses by using the singular change-of-variable-based construction. In our regularized setting the analogous statement is false: without the lossy layer, there are certain resonant inclusions (depending in general on ) that have a huge effect on the boundary measurements.

A new cloaking method is presented for 2D quasistatics and the 2D Helmholtz equation that we speculate extends to other linear wave equations. For 2D quasistatics it is proven how a single active exterior cloaking device can be used to shield an object from surrounding fields, yet produce very small scattered fields. The problem is reduced to finding a polynomial which is close to 1 in a disk and close to 0 in another disk, and such a polynomial is constructed. For the 2D Helmholtz equation it is numerically shown that three exterior cloaking devices placed around the object suffice to hide it.

Abstract:It is shown how a recently proposed method of cloaking is effective over a broad range of frequencies. The method is based on three or more active devices. The devices, while not radiating significantly, create a "quiet zone" between the devices where the wave amplitude is small. Objects placed within this region are virtually invisible. The cloaking is demonstrated by simulations with a broadband incident pulse. B 49(12), 8479-8482 (1994). 23. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). 24. O. P. Bruno and S. Lintner, "Superlens-cloaking of small dielectric bodies in the quasistatic regime," J. Appl. Phys. 102, 124,502 (2007)

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In this paper we extend the results proposed in [36] and study the problem of active control in the context of a scalar Helmholtz equation. Given a source region Da and {v 0 , v 1 ,. .. , vn}, a set of solutions of the homogeneous scalar Helmholtz equation in n mutually disjoint "control" regions {D 0 , D 1 ,. .. , Dn} of R 2 or R 3 , respectively, the main objective of this paper is to characterize the necessary boundary data on ∂Da so that the solution to the corresponding exterior scalar Helmholtz problem will closely approximate v i in D i , respectively, for each i ∈ {0,. .. , n}. Building up on the previous ideas presented in [36] we show the existence of a class of solutions to the problem, provide numerical support of the results in 2D and 3D and discuss the existence of a minimal energy solution and its stability. 1. Introduction. During recent years, there has been a growing interest in the development of feasible strategies for the control of acoustic and electromagnetic fields with one possible application being the construction of robust schemes for sonar or radar cloaking. One main approach controls fields in the regions of interest by changing the material properties of the medium in certain surrounding regions [4, 5, 8, 13, 14, 15, 39] (and the references therein). Several alternative techniques are proposed in the literature (other than transformation optics strategies) such as: plasmonic designs (see [2] and the references therein), strategies based on anomalous resonance phenomena (see [31, 32, 33]), conformal mapping techniques (see [24, 25]), and complementary media strategies (see [23]). In this paper, we will study an approximate control problem for the exterior scalar Helmholtz equation, i.e., we will characterize the boundary control functions on the source ∂D a so that we achieve Keywords and phrases. Active manipulation, Helmholtz equation, layer potentials, integral equation, active exterior cloaking. The AFOSR supported this work under the 2013 YIP Award FA9550-13-1-0078.

We consider propagation of waves through a spatio-temporal doubly periodic material structure with rectangular microgeometry in one spatial dimension and time. Both spatial and temporal periods in this dynamic material are assumed to be the same order of magnitude. Mathematically the problem is governed by a standard wave equationWe consider a checkerboard microgeometry where variables cannot be separated. The rectangles in a space-time checkerboard are assumed filled with materials differing in the values of phase velocities k ρ but having equal wave impedance kρ. The difference between dynamic materials and classical static composites is that in the former case the design variables will also be time dependent. Within certain parameter ranges, the formation of distinct and stable limiting characteristic paths, i.e., limit cycles, was observed in [K.A. Lurie, S.L. Weekes, Wave propagation and energy exchange in a spatio-temporal material composite with rectangular microstructure, J. Math. Anal. Appl. 314 (2006) 286-310]; such paths attract neighboring characteristics after a few time periods. The average speed of propagation along the limit cycles remains the same throughout certain ranges of structural parameters, and this was called in [K.A. Lurie, S.L. Weekes, Wave propagation and energy exchange in a spatiotemporal material composite with rectangular microstructure, J. Math. Anal. Appl. 314 (2006) 286-310] a plateau effect. Based on numerical evidence, it was conjectured in [K.A. Lurie, S.L. Weekes, Wave propagation and energy exchange in a spatio-temporal material composite with rectangular microstructure, J. Math. Anal. Appl. 314 (2006) 286-310] that a checkerboard structure is on a plateau if and only if it yields stable limit cycles and that there may be energy concentrations over certain time intervals depending on material parameters. In the present work we give a more detailed analytic characterization of these phenomena and provide a set of sufficient conditions for the energy concentration that was predicted numerically in [K.A. Lurie, S.L. Weekes, Wave propagation and energy exchange in a spatio-temporal material composite with rectangular microstructure, J. Math. Anal. Appl. 314 (2006) 286-310].

We cloak a region from a known incident wave by surrounding the region with three or more devices that cancel out the field in the cloaked region without significantly radiating waves. Since very little waves reach scatterers within the cloaked region, the scattered field is small and the scatterers are for all practical purposes undetectable. The devices are multipolar point sources that can be determined from Green's formula and an addition theorem for Hankel functions. The cloaking devices are exterior to the cloaked region.

In this paper we present a strategy for the the synthesis of acoustic sources with controllable near fields in free space and finite depth homogeneous ocean environments. We first present the theoretical results at the basis of our discussion and then, to illustrate our findings we focus on the following three particular examples:1. acoustic source approximating a prescribed field pattern in a given bounded subregion of its near field. 2. acoustic source approximating different prescribed field patterns in given disjoint bounded near field sub-regions. 3. acoustic source approximating a prescribed back-propagating field in a given bounded near field sub-region while maintaining a very low far field signature.For each of these three examples, we discuss the optimization scheme used to approximate their solutions and support our claims through relevant numerical simulations.arXiv:1706.05233v1 [math.OC]